Abstract
Graphene exhibits extraordinary electronic and mechanical properties, and extremely high thermal conductivity. Being a very stable atomically thick membrane that can be suspended between two leads, graphene provides a perfect test platform for studying thermal conductivity in twodimensional systems, which is of primary importance for phonon transport in lowdimensional materials. Here we report experimental measurements and nonequilibrium molecular dynamics simulations of thermal conduction in suspended singlelayer graphene as a function of both temperature and sample length. Interestingly and in contrast to bulk materials, at 300 K, thermal conductivity keeps increasing and remains logarithmically divergent with sample length even for sample lengths much larger than the average phonon mean free path. This result is a consequence of the twodimensional nature of phonons in graphene, and provides fundamental understanding of thermal transport in twodimensional materials.
Introduction
In bulk materials, heat conduction is governed by Fourier’s law: J=−κ∇T, where J is the local heat flux, ∇T is the temperature gradient and κ is thermal conductivity, which is an intrinsic parameter of the materials and is usually size and geometry independent. Over the past 200 years, Fourier’s law has been successfully employed to describe thermal conduction in threedimensional systems. However, with the recent availability of lowdimensional materials, it remains to verify whether thermal conductivity is still size/geometry independent. For onedimensional (1D) lattice models^{1,2,3,4,5} and quasi1D nanostructures and polymers^{6,7,8}, a length dependence of κ as~L^{β} (L is sample length) has been predicted theoretically and then experimentally verified for nanotubes^{9}. The same question remains open for twodimensional (2D) systems. While logarithmic divergence of the thermal conductivity has been demonstrated numerically for 2D lattice models^{2,3,10,11,12,13}, simulations of bidimensional materials such as graphene, give contradictory results, which depend on the employed simulation techniques and approximations^{14,15,16,17}. Graphene is believed to be a perfect test bed to investigate this size effect of 2D systems owing to its novel thermal properties^{18,19,20,22,24,25,26,27,28,29,30,31,32,33}. However, besides a recent experiment carried out on supported graphene nanoribbons^{34}, in which size effect, especially the length effect, is affected by graphene–substrate interaction^{35}, a compelling experimental investigation of lengthdependent thermal conductivity in suspended singlelayer graphene (SLG) is still missing, probably owing to the challenges in suspending this atomically thick membrane suitable for sizedependent thermal measurements.
Here we report measurements of thermal conduction in suspended SLG grown by chemical vapour deposition on copper (CuCVD)^{36,37,38}. Both the experimental and simulation results show that thermal conductivity scale with length as κ~logL at room temperature even when L is one order of magnitude larger than the average phonon mean free path (MFP), which we think is related to the 2D nature of phonons in graphene. Possible mechanisms of this divergent behaviour are discussed.
Results
Temperaturedependent thermal conduction
The experimentally measured thermal conductance σ=1/R_{total} and κ for all junctions are shown in Fig. 1c,d. Here κ is obtained from , in which R_{total} is the total measured thermal resistance, R_{c} is the thermal contact resistance and A is the unit crosssection area (we define A=w × h, where w is the width of the samples and h≈0.34 nm is the nominal thickness of SLG). Independently on the sample length, all samples exhibit qualitatively the same behaviour as a function of the temperature: σ(T) increases with temperature and reaches a broad plateau. The thermal conductivity at T=300 K in our longest sample (L=9 μm) shows a value of (1,689±100) Wm^{−1} K^{−1}~(1,813±111) Wm^{−1} K^{−1} (see Supplementary Note 1; Supplementary Tables 1 and 2 for the error bars), which is comparable to the values obtained from Ramanbased measurement techniques (from ~1,800 Wm^{−1} K^{−1} to ~5,300 Wm^{−1} K^{−1}, see Supplementary Note 2; Supplementary Table 3). Remarkably, our measurements show that κ increases with sample length over the entire measured temperature range.
Quasiballistic phonon transport in submicron samples
Here plotting σ/A is more instructive than κ. In all submicron samples, the thermal conductance exhibits the same temperature dependence (Fig. 2a). This has important implications for the nature of 2D phonon transport. It has been reported that phonons in graphene can travel without scattering, that is, ballistic transport^{34,39}. In clean devices, σ/A has been expected to follow Wm^{−2} K^{−1}, as shown by the black dashed curve in the same figure^{34}. Indeed, the experimentally measured values are not only length independent, but also within 39.8% of the predicted ballistic thermal conductance at T=30 K (phonons in quasiballistic regime at low temperature), indicating high sample quality and clean surface in the measured graphene.
Atomistic simulations provide a powerful tool for validation and interpretation of experimental measurements, with the advantage that the limit of perfectly crystalline samples can be probed. We have performed nonequilibrium molecular dynamics (NEMD)^{40} simulations of heat transport in graphene at similar conditions as the experiments. We considered periodic models of defectfree, isotopically pure and suspended graphene patches with sizes between 5.5 nm and 1.4 μm, at 300 K and at 1,000 K. Our simulations show that at 300 K, the thermal conductance remains constant for a length (between the hot and the cold reservoir) up to ~80 nm (σ/σ_{0}=0.84 at ~80 nm, where σ_{0} is the calculated thermal conductance in a 5.5nmlong sample), indicating ballistic heat transport in this size range (Fig. 2b).
Lengthdependent thermal conductivity
Before we discuss the length dependence of thermal transport at room temperature, we need to comment on the issue of thermal contact resistance R_{c}, a critical parameter in thermal conductivity studies in general. Note that while achieving low values for R_{c} with 1D nanowires and nanotubes remains difficult, the inherently larger contact area with 2D sheets makes it less challenging. Our calculations give a relatively small value of R_{c} with values ranging from ~9.9 × 10^{4} K W^{−1} to ~1.26 × 10^{6} K W^{−1}, which contributes ~0.9% to ~11.5% of R_{total} in the sample with 9 μm length (see Supplementary Note 3; Supplementary Table 4).
Therefore, we plot κ versus L in Fig. 3a for different thermal contact resistance, that is, negligible R_{c}, R_{c}=5.5 × 10^{5} K W^{−1} (~5% of R_{total} in 9 μm sample), and R_{c}=1.26 × 10^{6} K W^{−1} (~11.5% of R_{total} in 9 μm sample), respectively. κ increases with sample length at T=300 K. Surprisingly and in contrast with experiments on supported graphene^{34}, κ does not saturate even in our longest sample with L=9 μm. More importantly, κ scales with ~logL in samples with L≥700 nm (we do not claim the ~logL behaviour in shorter samples owing to the large effects from R_{c}). Such behaviour has been predicted for 2D systems and is a direct consequence of the 2D nature of phonons^{2,3}. This anomalous lengthdependent thermal conductivity should not be confused with quasiballistic transport discussed above, which occurs when the length of the sample is comparable to the average phonon MFP of the main heat carriers, that is, acoustic phonons in graphene. Following the parameters used by Ghosh et al.^{28}, we estimate the average MFP to be around 240 nm at T=300 K. Please note that MFP is a spectral property that depends on frequency and polarization of each phonon, therefore MFP=240 nm is an average or effective MFP of all the phonons contributing to thermal conduction. Moreover, our simulation results show that the thermal conductance starts decreasing for length between 43 nm and 80 nm (triangles in Fig. 2b) and is σ/σ_{0}~0.6 for L~171 nm. Therefore, L>MFP holds in our shortest samples at T=300 K and κ does not saturate even when the length of the sample is >30 times larger than the average MFP.
It is worth pointing out that, in contrast with 2D lattice models^{10,13}, graphene atoms can move out of plane, giving rise to flexural acoustic (ZA) and flexural optical (ZO) modes. The role of the ZA modes of graphene in heat transport is still debated: according to lattice dynamics calculations and equilibrium molecular dynamics (MD) simulations, the ZA modes provide the fundamental scattering channels to limit κ to a finite value, unless strain is applied^{16,17}. On the other hand calculations that take into account the deviations of phonon populations from equilibrium, through a selfconsistent solution of the Peierls–Boltzmann equation, suggest that ZA modes are the main heat carriers in graphene, and show a significant size dependence of κ at the micrometre scale^{41}. Here we have probed direct stationary nonequilibrium heat transport at conditions very similar to the experiments by NEMD simulations. In sharp contrast with previous equilibrium MD simulations, but in excellent agreement with experiments, we observe κ(L)~logL both at T=300 K and at 1,000 K, for systems with size well beyond the ballistic length (Fig. 3b). Surprisingly, in spite of the unavoidable presence of boundary scattering at the interface with the thermal reservoirs, κ from NEMD exceeds the converged value of κ from equilibrium Green–Kubo calculations, computed with the same empirical potential, indicating that the mechanism of heat transport is substantially different at stationary nonequilibrium. We also note that different NEMD approaches, namely reverse NEMD^{40} and direct NEMD^{42}, give the same result at any given size. In addition for each L, κ(L) is not sensitive to the magnitude of the applied heat flux or temperature gradients. In our NEMD simulations, we ensured that grad(T) is proportional to J, meaning that we do not break the linear response regime (see Supplementary Note 4; Supplementary Fig. 1). We have also carefully considered the effect of different aspect ratios of the simulation cell, probing very large systems of square aspect ratio, which give results in agreement with those obtained with elongated supercells (Fig. 3b), and confirm that the log(L) divergence is not an artifact of the simulation setup.
Discussion
We now seek to understand the physics behind the observed lengthdependent thermal conductivity. A number of theoretical and computational studies on ideal systems, such as the Fermi–Pasta–Ulam model, have demonstrated that κ in lowdimensional systems depends on system size (inset of Fig. 3a). In fact, rigorous mathematical proof tells us that for a momentumconserved 1D system, κ diverges with system size^{1,2,3,4}, which has been confirmed experimentally in nanotubes^{9}. On the other hand, the length dependence of 2D systems remains unresolved and a ~logL dependence has been proposed by various analytical theories^{2,3,10,13}. Our experiment and simulation studies provide strong evidence for a ~logL behaviour in graphene, which is different from both of that in 1D and threedimensional systems.
The other possible mechanism behind the lengthdependent thermal conductivity is related to the ballistic propagation of extremely longwavelength, lowfrequency acoustic phonons^{29,43}. Nika et al.^{29} emphasized the importance of lowfrequency acoustic phonons in graphene. As sample size increases, more lowfrequency acoustic phonons can be excited and contribute to thermal conduction, resulting in a lengthdependent behaviour. The third possibility is related to the selection rules for threephonon scattering, the phase space of which is strongly restricted by the reduced dimensionality^{41,44}; however, a nonlogarithmic dependence of the thermal conductivity was reported when threephonon process was considered to second order.
A comparative analysis of phonon populations in MD simulations at equilibrium and nonequilibrium conditions suggests that in the latter conditions the population of outofplane modes is augmented, whereas inplane modes with polarization in the direction of the heat flux propagation get slightly depopulated (see Supplementary Fig. 2). Such population imbalance produces the observed discrepancy between equilibrium and nonequilibrium simulations, promoting logarithmic divergence of κ at stationary nonequilibrium (see Supplementary Note 4 and Methods for further details). Our simulations therefore suggest that logarithmic divergence stems from the combination of reduced dimensionality and displacements of phonon populations at stationary nonequilibrium conditions.
From experiments, it is still challenging to distinguish which transport mechanism dominates the observed lengthdependent thermal conductivity. To this end, it would be constructive to show the relative contribution to thermal conductivity from phonons with different frequencies, or to study the thermal conductivity in samples with much larger size (for example, 100 μm or even millimetre), which would however require the development of new measurement techniques.
In summary, we have studied thermal conductivity in suspended SLG. Thermal conductivity has been observed to increase with the length of the samples and to scale as ~logL, even when L is one order of magnitude longer than the average phonon MFP. Possible mechanisms have been discussed: MD simulations, in excellent agreement with experiments, confirm the divergence of κ and suggest that it is related to the 2D nature of phonons in graphene, and to the change of the phonon population at stationary nonequilibrium conditions.
Methods
MD simulations
NEMD simulations were performed using LAMMPS ( http://lammps.sandia.gov/). Interatomic forces were described by the Tersoff potential with a parameter set optimized for graphene^{45}. Simulations employed periodic boundary conditions in the graphene plane, and each supercell was relaxed at the simulation temperature to achieve zero inplane stress. Two different NEMD simulation methods were used, in which the thermal conductivity is calculated directly from the temperature gradient and the heat flux via Fourier’s law. The simulation supercell is divided in contiguous slabs along the direction of heat propagation (see Supplementary Fig. 3). Each slab contains ~440 atoms and the temperature of each slab is calculated from its average kinetic energy (K_{E}) as T=2/3 K_{E}/k_{B}. The two methods differ as for the mechanism employed to generate a stationary heat flux. In the direct NEMD method^{42}, independent Langevin thermostats were used to control the temperature of the cold and hot slabs. In the reverse nonequilibrium MD method^{31} (RNEMD) the temperature gradient is imposed by swapping the velocity of the slowest molecule in the cold slab with the velocity of the fastest one in the hot slab. After a transient period, a stationary temperature gradient is established in the system due to the imposed heat flux. However, the velocity exchange disturbs the stability of the numerical integration of the equations of motion and introduces a drift in the total energy of the system. Velocities were exchanged every 200 steps (20 fs), and we used a timestep of 0.1 fs for a stable integration of the equations of motion. Each NEMD simulation ran for at least 40 million steps, and the temperature gradient and heat flux were taken as the average over the final 10 million steps. Error bars are the s.d. of the averaged data. Supplementary Fig. 4 shows the temperature profile for two supercell sizes, using both NEMD and RNEMD methods. Temperature gradients are taken as the average slope considering both sides of the simulation cell, only in the linear regions between cold and hot layers. Such temperature gradients, although larger in RNEMD simulations than in NEMD, are comparable to the ones in the experiment.
Initially, the simulation cells were increased in length while keeping a constant width of ~11 nm. This procedure implies an increase in the aspect ratio of the simulation cells from a ratio of 1 for the smallest cell (4,400 atoms), up to a ratio of ~250 for the longest cell (1.1 million atoms). In order to assure that the length dependence observed in our results was not related to the increase in aspect ratio, we also performed simulations with square cells up to ~560 nm (11.5 million atoms), which are in agreement with the observed log divergence as shown in Fig. 3b.
Validation tests of the simulation methods are described in Supplementary Note 4; Supplementary Fig. 1.
Fabrication details
We employed typical prepatterned heater wires^{46,47} (see Fig. 1a,b) to measure thermal conductance of suspended SLG grown by CuCVD^{36,38}. Our device fabrication starts with SiN_{x} membranebased heater structures (Supplementary Fig. 5a) optimized for lengthdependent studies. To obtain highquality samples, the CVD graphene was grown by reusing the copper catalyst^{37}, which in general results in better transport properties. In order to decrease the thermal contact resistance, SiN_{x} wafer and Pt prepatterned electrodes were cleaned for an extended period of time (>10 min) in O_{2} plasma, followed by immediate CVD graphene transfer. This helps to clean organic residue on the Pt electrodes and create suspended chemical bonds, resulting in a further improvement of the thermal contact to graphene^{48}. The graphene sheets were then patterned by standard electron beam lithography, followed by an O_{2} plasma step (see Supplementary Fig. 5c). In a second electron beam lithography step, 30 nm Cr/Au bars were deposited on both ends of the graphene stripes to ensure good thermal contact with the Pt electrodes underneath graphene (see Supplementary Fig. 5d). After suspending graphene by wet etching, the devices were dried using a critical point dryer to avoid damage due to surface tension. It is worth noting that we do not etch away all the silicon under SiN_{x} pad (that is, heater and sensor), but left few micrometres to few tens of micrometresthick silicon to act as a massive heat batch and make sure that heat spreads uniformly within the SiN_{x} pad during thermal measurements. Graphene samples were annealed for 5 h at 300 °C in flowing hydrogen (H_{2}: 150 ml min^{−1}, Ar: 150 ml min^{−1}, 1 atm) to remove possible organic residues before measurements.
Sample geometry
The length dependence of thermal conductivity is studied by varying the channel length from 300 nm to 9 μm. Since the thermal conductivity is also weakly width (w) dependent when the width is larger than 1.5 μm (see Supplementary Fig. 6; Supplementary Note 5), we have kept the width fixed at 1.5 μm. We define the direction of heat flow as length, and the perpendicular direction as width.
Thermal conductivity measurement
The suspended graphene sheet provides a thermal path between the two SiN_{x} membranes that are otherwise thermally and electrically isolated from each other (Fig. 1a). A μADC current combined with an AC current (100–200 nA) was applied to the heater resistor (R_{h}, red Pt coil in Fig. 1a). The DC current was used to apply Joule heat in R_{h} and to increase its temperature (T_{h}) from the environment temperature, T_{0}. The AC current was used to measure the resistance of R_{h}. The Joule heating in R_{h} gradually dissipates through the six Pt/SiN_{x} beams and the graphene sheet, which raises the temperature (T_{s}) in the sensor resistor (R_{s}, blue Pt coil in Fig. 1a). In the steady state, the thermal conductance of the graphene sheet, σ_{G} and each of the suspended Pt/SiN_{x} beam, σ_{l}, can be obtained from
and
where ΔT_{h} and ΔT_{s} indicate the temperature rise in the R_{h} and R_{s}, Q_{h} and Q_{l} indicate the Joule heat applied to the heater R_{h} and the Pt leads. The temperature change in each membrane as function of applied Joule heat was kept in the linear range. The increase in temperature, ΔT_{h} and ΔT_{s}, was controlled to below 5 K to minimize thermal radiation between the two membranes. All measurements were performed under vacuum conditions better than 1 × 10^{−6} mbar.
Additional information
How to cite this article: Xu, X. et al. Lengthdependent thermal conductivity in suspended singlelayer graphene. Nat. Commun. 5:3689 doi: 10.1038/ncomms4689 (2014).
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Acknowledgements
This work was supported in part by Singapore National Research Foundation Fellowship (NRFRF200807), NRFCRP grant (R143000360281), NRFCRP grant (R144000295281), CRP Award No. NRFCRP920113, by MOE T2 Grant R144000305112, by A*STAR grant (R143000360311), by NUS NanoCore, and by the Singapore Millennium Foundation SMF/NUS Horizon grant (R144000271592/646). L.F.C.P. and D.D. acknowledge the provision of computational facilities and support by Rechenzentrum Garching of the Max Planck society (MPG), and access to the supercomputer JUGENE at the Ju¨lich Supercomputing Centre under NIC project HMZ26. Financial support was provided by MPG under the MPRG program.
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X.X. and B.Ö. designed the experiment, X.X. performed experiments, X.X., B.L. and B.Ö. analysed data and wrote the paper; L.F.C.P. and D.D. designed and carried out the simulations and wrote the paper; Y.W., S.B., B.H.H. and K.P.L. grew CVD graphene; J.W., K.Z. and X.Z. helped prepare and measure samples; C.T.B., R.X. and J.T.L.T. prepared the prepatterned Pt/SiN_{x} wafers; B.L. and B.Ö. directed the project.
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Xu, X., Pereira, L., Wang, Y. et al. Lengthdependent thermal conductivity in suspended singlelayer graphene. Nat Commun 5, 3689 (2014). https://doi.org/10.1038/ncomms4689
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DOI: https://doi.org/10.1038/ncomms4689
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